In: Statistics and Probability
Please include all steps. Thanks
An urn model is known in the field of probability and statistics as a useful representation of a
probabilistic problem using coloured balls in an urn. There are many different variations of an
urn model. For this problem, consider the following case:
• There is an urn with 1 red ball and 1 blue ball.
• Every time a ball is drawn (at random) from the urn, it is placed back in the urn along with
2 more balls of the same colour as the ball that was drawn, and 1 more ball of the other
colour.
Denote the random variable Xi to be the number of red balls after the i-th drawn ball (for
i = 1, 2 . . .). Note that Xi is the random variable for the number of red balls in the urn including
the three new balls added after the i-th draw.
(a) Find the probability mass function (pmf) of X2.
(b) What is the probability that the first ball drawn was red, given that there are at least 5 red
balls after the third ball is drawn.
(c) Compute E(X3) and Var(X3).